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\title{Complex Analysis}
\subtitle{Chapter 3. Analytic Functions as Mappings \\ Section 2. Conformality }
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] Elementary Point Set Topology
\begin{enumerate}
\item[1.1.] Sets and Elements
\item[1.2.] Metric Spaces
\item[1.3.] Connectedness
\item[1.4.] Compactness
\item[1.5.] Continuous Functions
\item[1.6.] Topological Spaces
\end{enumerate}

\item[2.] {\color{red}Conformality}
\begin{enumerate}
\item[2.1.] {\color{red}Arcs and Closed Curves}
\item[2.2.] {\color{red}Analytic Functions in Regions}
\item[2.3.] {\color{red}Conformal Mapping}
\item[2.4.] {\color{red}Length and Area}
\end{enumerate}

\end{enumerate}

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\begin{enumerate}
\item[3.] Linear Transformations
\begin{enumerate}
\item[3.1.] The Linear Group
\item[3.2.] The Cross Ratio
\item[3.3.] Symmetry
\item[3.4.] Oriented Circles
\item[3.5.] Families of Circles
\end{enumerate}

\item[4.] Elementary Conformal Mappings
\begin{enumerate}
\item[4.1.] The Use of Level Curves
\item[4.2.] A Survey of Elementary Mappings
\item[4.3.] Elementary Riemann Surfaces
\end{enumerate}
 
\end{enumerate}

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\begin{frame}{2. Conformality }

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\begin{enumerate}

\item[1.] 
We now return to our original setting where all functions and variables are restricted to real or complex numbers. 

\item[2.] 
The role of metric spaces will seem disproportionately small: all we actually need are some simple applications of {\color{blue}connectedness} and {\color{blue}compactness}. 


\item[3.] 
The whole section is mainly descriptive. 

\item[4.] 
{\color{red}It centers on the geometric consequences of the existence of a derivative.}

\end{enumerate}

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\begin{enumerate}

\item[1.] 
The equation of an arc $\gamma$ in the plane is most conveniently given in parametric form $x = x(t)$, $y = y(t)$ where $t$ runs through an interval $\alpha\le t \le \beta$ and $x(t)$, $y(t)$ are continuous functions.

\item[2.]  
We can also use the complex notation $z = z(t) = x(t) + iy(t)$ which has several advantages. 

\item[3.]  
It is also customary to identify the arc $\gamma$ with the continuous mapping of $[\alpha,\beta]$.

\item[4.]  
When following this custom it is preferable to denote the mapping by $z = \gamma(t)$.


\end{enumerate}

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\begin{enumerate}


\item[5.]  
Considered as a point set an arc is the image of a closed finite interval under a continuous mapping.

\item[6.]  
As such it is {\color{blue}compact} and {\color{blue}connected}.

\item[7.]  
{\color{red}However, an arc is not merely a set of points, but very essentially also a succession of points, ordered by increasing values of the parameter. 
}

\item[8.]  
If a nondecreasing function $t = \varphi(\tau)$ maps an interval $\alpha' \le \tau \le \beta'$ onto $\alpha \le t \le \beta$, then $z = z(\varphi(\tau))$ defines the same succession of points as $z = z(t)$.

\end{enumerate}

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\begin{enumerate}


\item[9.]  
We say that the first equation arises from the second by a change of parameter.

\item[10.]  
The change is reversible if and only if $\varphi(\tau)$ is strictly increasing. 

\item[11.]  
For instance, the equation $z = t^2 + it^4, 0\le t \le 1$ arises by a reversible change of parameter from the equation $z = t + it^2, 0 \le t \le 1$.

\item[12.]  
A change of the parametric interval $(\alpha,\beta)$ can always be brought about by a linear change of parameter, which is one of the form $t = a\tau + b, a > 0$.


\end{enumerate}

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\begin{enumerate}



\item[13.]  
Logically, the simplest course is to consider two arcs as different as soon as they are given by different equations, regardless of whether one equation may arise from the other by a change of parameter. 

\item[14.]  
{\color{red}In following this course, as we will, it is important to show that certain properties of arcs are invariant under a change of parameter. 
}

\item[15.]  
For instance, the initial and terminal point of an arc remain the same after a change of parameter. 


\end{enumerate}

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\begin{enumerate}



\item[16.]  
{\color{red}If the derivative $z'(t) = x'(t) + iy'(t)$ exists and is $\neq 0$, the arc $\gamma$ has a tangent whose direction is determined by $\mathrm{arg}\, z'(t)$. 
}

\item[17.]  
We shall say that the arc is {\color{blue}differentiable} if $z'(t)$ exists and is continuous (the term continuously differentiable is too {\color{blue}unwieldy}); if, in addition, $z'(t) \neq 0$ the arc is said to be {\color{blue}regular}.

\item[18.]  
An arc is {\color{blue}piecewise differentiable} or {\color{blue}piecewise regular} if the same conditions hold except for a finite number of values $t$; at these points $z(t)$ shall still be continuous with left and right derivatives which are equal to the left and right limits of $z'(t)$ and, in the case of a piecewise regular arc, $\neq 0$.

\end{enumerate}

\hfill 

\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

{\footnotesize unwieldy: difficult to carry or move because of its size, shape, or weight}



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\begin{enumerate}

\item[19.]  
The differentiable or regular character of an arc is invariant under the change of parameter $t = \varphi(\tau)$ provided that $\varphi'(\tau)$ is continuous and, for
regularity, $\neq 0$. 

\item[20.]  
When this is the case, we speak of a differentiable or regular change of parameter.

\item[21.]  
An arc is simple, or a {\color{blue}Jordan arc}, if $z(t_1) = z(t_2)$ only for $t_1 = t_2$.

\item[22.]  
An arc is a {\color{blue}closed curve} if the end points coincide: $z(\alpha) = z(\beta)$.


\end{enumerate}

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\begin{enumerate}

\item[23.]  
For closed curves {\color{blue}a shift of the parameter} is defined as follows. 

\item[24.]  
If the original equation is $z = z(t), \alpha \le t \le \beta$, we choose a point $t_0$ from the interval $(\alpha,\beta)$ and define a new closed curve whose equation is $z = z(t)$ for $t_0 \le t \le \beta$ and $z = z(t -\beta + \alpha)$ for $\beta \le t \le t_0 + \beta - \alpha$. 

\item[25.]  
{\color{red}The purpose of the shift is to get rid of the distinguished position of the initial point. }

\item[26.]  
The correct definitions of a differentiable or regular closed curve and of a simple closed curve (or Jordan curve) are obvious.

\end{enumerate}

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\begin{enumerate}

\item[27.]  
The opposite arc of $z = z(t), \alpha \le t \le \beta$, is the arc $z = z(-t), -\beta \le t \le -\alpha$. 

\item[28.]  
Opposite arcs are sometimes denoted by $\gamma$ and $-\gamma$, sometimes by $\gamma$ and $\gamma^{-1}$ depending on the connection. 

\item[29.]  
A constant function $z(t)$ defines a point curve. 

\end{enumerate}

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\begin{enumerate}

\item[30.]  
A circle $C$, originally defined as a locus $|z - a| = r$, can be considered as a closed curve with the equation $z = a+ re^{it}, 0 \le t \le 2\pi$.

\item[31.]  
We will use this standard parametrization whenever a circle is introduced. 

\item[32.]  
This convention saves us from writing down the equation each time it is needed; also, and this is its most important purpose, it serves as a definite rule to distinguish between $C$ and $-C$.


\end{enumerate}

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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
When we consider the derivative
\begin{equation*}
f'(z) = \lim\limits_{h\to 0} \frac{f(z+h)-f(z)}{h}
\end{equation*}
of a complex-valued function, defined on a set A in the complex plane, it is of course understood that $z\in A$ and that the limit is with respect to values $h$  such that $z + h \in A$. 

\item[2.] 
The existence of the derivative will therefore have a different meaning depending on whether $z$ is an interior point or a boundary point of $A$. 

\item[3.] 
{\color{red}The way to avoid this is to insist that all analytic functions be defined on open sets. 
}

\end{enumerate}

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\begin{enumerate}

\item[4.] 
We give a formal statement of the definition.


\item[5.] 
{\color{red}Definition 10. A complex-valued function $f(z)$, defined on an {\color{blue}open set} $\Omega$, is said to be {\color{blue}analytic} in $\Omega$ if it has a derivative at each point of $\Omega$. }


\item[6.] 
Sometimes one says more explicitly that $f(z)$ is {\color{blue}complex analytic}. 

\item[7.] 
A commonly used synonym is {\color{blue}holomorphic}.

\item[8.] 
It is important to stress that the open set $\Omega$ is part of the definition. 

\item[9.] 
As a rule one should avoid speaking of an analytic function $f(z)$ without referring to a specific open set $\Omega$ on which it is defined, but the rule can be broken if it is clear from the context what the set is.


\end{enumerate}

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\begin{enumerate}

\item[10.] 
Observe that $f$ must first of all be a function, and hence single-valued. 

\item[11.] 
If $\Omega'$ is an open subset of $\Omega$, and if $f(z)$ is analytic in $\Omega$, then the restriction of $f$ to $\Omega'$ is analytic in $\Omega'$; it is customary to denote the restriction by the same letter $f$.

\item[12.] 
In particular, since the components of an open set are open, it is no loss of generality to consider only the case where $\Omega$ is connected, that is to say a {\color{blue}region}.

\end{enumerate}

\vfill

\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

{\footnotesize region: an open and connected subset of the complex plane}


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\begin{enumerate}


\item[13.] 
For greater flexibility of the language it is desirable to introduce the following complement to Definition 10.

\item[14.] 
{\color{red}Definition 11. A function $f(z)$ is {\color{blue}analytic on an arbitrary point set} $A$ if it is the restriction to $A$ of a function which is analytic in some open set containing $A$. }


\item[15.] 
The last definition is merely an agreement to use a convenient terminology.

\item[16.] 
This is a case in which the set $\Omega$ need not be explicitly mentioned, for the specific choice of $\Omega$ is usually {\color{blue}immaterial} as long as it contains $A$. 

\end{enumerate}

\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

{\footnotesize immaterial: unimportant under the circumstances; irrelevant}


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\begin{enumerate}

\item[17.] 
Another instance in which the mention of $\Omega$ can be suppressed is the phrase: ``Let $f(z)$ be analytic at $z_0$.'' 

\item[18.] 
It means that a function $f(z)$ is defined and has a derivative in some unspecified open neighborhood of $z_0$. 

\item[19.] 
Although our definition requires all analytic functions to be single-valued, it is possible to consider such multiple-valued functions as $\sqrt{z}$, $\log z$, or $\arccos z$, provided that they are restricted to a definite region {\color{red}in which it is possible to select a single-valued and analytic branch of the function. 
}

\end{enumerate}

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\begin{enumerate}


\item[20.] 
For instance, {\color{red}we may choose for $\Omega$ the complement of the negative real axis $z \le 0$; this set is indeed open and connected. 
}

\item[21.] 
{\color{red}In $\Omega$ one and only one of the values of $\sqrt{z}$ has a positive real part.
}

\item[22.] 
{\color{red}With this choice $w = \sqrt{z}$ becomes a single-valued function in $\Omega$; let us prove that it is continuous. 
}

\item[23.] 
Choose two points $z_1,z_2\in \Omega$ and denote the corresponding values of $w$ by $w_1 = u_1+iv_1$, $w_2 = u_2 + iv_2$ with $u_1, u_2 > 0$. 

\end{enumerate}

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\begin{enumerate}


\item[24.] 
Then
\begin{equation*}
|z_1 - z_2| = |w_1^2 - w_2^2| = |w_1-w_2|\cdot |w_1+w_2|
\end{equation*}
and $|w_1+w_2|\ge u_1+u_2 >u_1$. 

\item[25.] 
Hence
\begin{equation*}
|w_1 - w_2| < \frac{|z_1-z_2|}{u_1}
\end{equation*}
and it follows that $w = \sqrt{z}$ is continuous at $z_1$.

\end{enumerate}

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\begin{enumerate}


\item[26.] 
{\color{red}Once the continuity is established the analyticity follows by derivation of the inverse function $z = w^2$. 
}

\item[27.] 
Indeed, with the notations used in calculus $\Delta z \to 0$ implies $\Delta w \to 0$. 

\item[28.] 
Therefore,
\begin{equation*}
\frac{dw}{dz} = \frac{1}{\frac{dz}{dw}} = \frac{1}{2w} = \frac{1}{2\sqrt{z}}
\end{equation*}
and we obtain with the same branch of $\sqrt{z}$.


\end{enumerate}

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\begin{enumerate}

\item[29.] 
{\color{red}In the case of $\log z$ we can use the same region $\Omega$, obtained by excluding the negative real axis, and define the principal branch of the logarithm by the condition $|\mathrm{Im}\,\log z| < \pi$.
}

\item[30.] 
Again, the continuity must be proved, but this time we have no algebraic identity at our disposal, and we are forced to use a more general reasoning. 

\item[31.] 
Denote the principal branch by $w = u + iv = \log z$.

\item[32.] 
For a given point $w_1 = u_1 + iv_1, |v_1| < \pi$, and a given $\varepsilon > 0$, consider the set $A$ in the $w$-plane which is defined by the inequalities $|w -
w_1| \ge \varepsilon$, $|v| \le \pi$, $|u - u_1| \le \log 2$.


\end{enumerate}

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\item[33.] 
This set is closed and bounded, and for sufficiently small $\varepsilon$ it is not empty. 

\item[34.] 
The continuous function $|e^w -e^{w_1}|$ has consequently a minimum $\rho$ on $A$ (Theorem 8, Corollary). 

\item[35.] 
This minimum is positive, for $A$ does not contain any point $w_1 + n\cdot 2\pi i$.

\item[36.] 
Choose $\delta = \min (\rho,\frac{1}{2}e^{u_1})$, and assume that 
$$|z_1 - z_2| = |e^{w_1} - e^{w_2}| < \delta. $$

\item[37.] 
Then $w_2$ cannot lie in $A$, for this would make $|e^{w_1} - e^{w_2}| \ge \rho \ge \delta$.

\end{enumerate}

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\item[38.] 
Neither is it possible that $u_2 < u_1- \log 2$ or $u_2 > u_1 + \log 2$; in the former case we would obtain $|e^{w_1} - e^{w_2}| \ge e^{u_2} - e^{u_1} > \frac{1}{2}e^{u_1} \ge \delta$, and in the latter case $|e^{w_1} - e^{w_2}| \ge e^{u_2} -e^{u_1} > e^{u_1} > \delta$.

\item[39.] 
Hence $w_2$ must lie in the disk $|w - w_1| < \varepsilon$, and we have proved that $w$ is a continuous function of $z$.

\item[40.] 
From the continuity we conclude as above that the derivative exists and equals $1/z$.

\end{enumerate}

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\begin{enumerate}

\item[41.] 
{\color{red}The infinitely many values of $\arccos z$ are the same as the values of $$\arccos z = i\log (z+\sqrt{z^2-1}).$$ 
}

\item[42.] 
{\color{red}In this case we restrict $z$ to the complement $\Omega'$ of the half lines $x \le -1, y = 0$ and $x \ge 1, y = 0$.
}

\item[43.] 
Since $1 -z^2$ is never real and $\le 0$ in $\Omega'$, we can define $\sqrt{1-z^2}$ as in the first example and then set $\sqrt{z^2-1} = i\sqrt{1 - z^2}$. 

\item[44.] 
Moreover, $z + \sqrt{z^2 -1}$ is never real in $\Omega'$, for $z + \sqrt{z^2-1}$ and $z-\sqrt{z^2-1}$ are reciprocals and hence real only if $z$ and $\sqrt{z^2-1}$ are both real; this happens only when $z$ lies on the excluded parts of the real axis. 

\end{enumerate}

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\item[45.] 
Because $\Omega'$ is connected, it follows that all values of $z + \sqrt{z^2-1}$ in $\Omega'$ are on the same side of the real axis, and since $i$ is such a value they are all in the upper half plane. 

\item[46.] 
We can therefore define an analytic branch of $\log (z + \sqrt{z^2 -1})$ whose imaginary part lies between $0$ and $\pi$.

\item[47.] 
In this way we obtain a single-valued analytic function
$$ \mathrm{arc cos}\, z = i \log (z + \sqrt{z^2 - 1})$$
in $\Omega'$ whose derivative is
$$D\, \mathrm{arc cos}\, z = i \frac{1}{z + \sqrt{z^2 - 1}}
\left( 1+ \frac{z}{\sqrt{z^2-1}} \right) = \frac{1}{\sqrt{1-z^2}}
$$
where $\sqrt{1-z^2}$ has a positive real part.


\end{enumerate}

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\item[48.] 
{\color{red}There is nothing unique about the way in which the region and the single-valued branches have been chosen in these examples.
}

\item[49.] 
Therefore, each time we consider a function such as $\log z$ the choice of the branch has to be specified. 

\item[50.] 
It is a fundamental fact that it is impossible to define a single-valued and analytic branch of $\log z$ in certain regions. 

\item[51.] 
This will be proved in the chapter on integration.

\item[52.] 
All the results of Chap. II, Sec. 1.2 remain valid for functions which are analytic on an open set.

\end{enumerate}

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\item[53.] 
In particular, the real and imaginary parts of an analytic function in $\Omega$ satisfy the Cauchy-Riemann equations
$$
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\,\,\,\,
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. 
$$

\item[54.] 
Conversely, if $u$ and $v$ satisfy these equations in $\Omega$, and if the partial
derivatives are continuous, then $u + iv$ is an analytic function in $\Omega$.

\item[55.] 
An analytic function in $\Omega$ degenerates if it reduces to a constant. 

\item[56.] 
In the following theorem we shall list some simple conditions which have this
consequence. 


\end{enumerate}

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\item[57.] 
{\color{red}Theorem 11. An analytic function in a region $\Omega$ whose derivative vanishes identically must reduce to a constant. The same is true if either the real part, the imaginary part, the modulus, or the argument is constant. }

 
\item[58.] 
The vanishing of the derivative implies that $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, $\partial v/\partial y$ are all zero. 

\item[59.] 
It follows that $u$ and $v$ are constant on any line segment in $\Omega$ which is parallel to one of the coordinate axes. 

\item[60.] 
In Sec. 1.3 we remarked, in connection with Theorem 3, that any two points in a region can be joined within the region by a polygon whose sides are parallel to
the axes. 

\item[61.] 
We conclude that $u + iv$ is constant. 


\end{enumerate}

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\item[62.] 
If $u$ or $v$ is constant,
$$f'(z) = \frac{\partial u}{\partial x} -i \frac{\partial u}{\partial y} 
= \frac{\partial v}{\partial y} + i \frac{\partial v}{\partial x} = 0,
$$
and hence $f(z)$ must be constant. 

\item[63.] 
If $u^2 + v^2$ is constant, we obtain
$$
u\frac{\partial u}{\partial x} + v \frac{\partial v}{\partial x} =0 
$$
and
$$
u\frac{\partial u}{\partial y} + v \frac{\partial v}{\partial y} 
= -u \frac{\partial v}{\partial x} + v \frac{\partial u}{\partial x} =0. 
$$

\end{enumerate}

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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}


\item[64.] 
These equations permit the conclusion 
$ \partial u/\partial x = \partial v/\partial x = 0$ 
unless the determinant $u^2 + v^2$ vanishes. 

\item[65.] 
But if $u^2 + v^2 = 0$ at a single point it is constantly zero and $f(z)$ vanishes identically.

\item[66.] 
Hence $f(z)$ is in any case a constant.

\item[67.] 
Finally, if $\mathrm{arg}\, f(z)$ is constant, we can set $u = kv$ with constant $k$
(unless $v$ is identically zero). 


\item[68.] 
But $u - kv$ is the real part of $(1+ik)f$, and we conclude again that $f$ must reduce to a constant. 

\item[69.] 
Note that for this theorem it is essential that $\Omega$ is a region. 

\item[70.] 
If not, we can only assert that $f(z)$ is constant on each component of $\Omega$.



\end{enumerate}

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\begin{frame}{2.2. Analytic Functions in Regions. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Give a precise definition of a single-valued branch of $\sqrt{1+z} + \sqrt{1-z}$ in a suitable region, and prove that it is analytic. 
}

\item  

\end{itemize}

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\begin{frame}{2.2. Analytic Functions in Regions. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Same problem for $\log \log z$.
}

\item  

\end{itemize}

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\begin{frame}{2.2. Analytic Functions in Regions. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Suppose that $f(z)$ is analytic and satisfies the condition $|f(z)^2-1|<1$ in a region $\Omega$. Show that either $\mathrm{Re}\, f(z) > 0$ or $\mathrm{Re}\, f(z) < 0$  throughout $\Omega$.
}

\item  

\end{itemize}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
Suppose that an arc $\gamma$ with the equation $z = z(t), \alpha\le t\le \beta$, is contained in a region $\Omega$, and let $f(z)$ be defined and continuous in $\Omega$.

\item[2.] 
Then the equation $w = w(t) = f(z(t))$ defines an arc $\gamma'$ in the $w$-plane which may be called {\color{blue}the image of $\gamma$}. 

\item[3.] 
Consider the case of an $f(z)$ which is analytic in $\Omega$. 

\item[4.] 
If $z'(t)$ exists, we find that $w'(t)$ also exists and is determined by 
\begin{equation}
w'(t) = f'(z(t)) z'(t). 
\label{eq-1}
\end{equation}

\item[5.] 
{\color{red}We will investigate the meaning of this equation at a point $z_0 = z(t_0)$ with $z'(t_0)\neq 0$ and $f'(z_0)\neq 0$. }

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[6.] 
The first conclusion is that $w'(t_0)\neq 0$. 

\item[7.] 
Hence $\gamma'$ has a tangent at $w_0 = f(z_0)$, and its direction is determined by 
\begin{equation}
\mathrm{arg}\, w'(t_0) = \mathrm{arg}\, f'(z_0) + \mathrm{arg}\, z'(t_0).
\label{eq-2}
\end{equation}

\item[8.] 
{\color{red}This relation asserts that the angle between the directed tangents to $\gamma$ at $z_0$ and to $\gamma'$ at $w_0$ is equal to $\mathrm{arg}\, f'(z_0)$. 
}

\item[9.] 
It is hence independent of the curve $\gamma$. 

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. \hfill 作业3B-2a }

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\begin{enumerate}


\item[10.] 
For this reason curves through $z_0$ which are tangent to each other are mapped onto curves with a common tangent at $w_0$. 

\item[11.] 
{\color{red}Moreover, two curves which form an angle at $z_0$ are mapped upon curves forming the same angle, in {\color{blue}sense} as well as in {\color{blue}size}. 
}

\item[12.] 
In view of this property the mapping by $w = f(z)$ is said to be {\color{blue}conformal} at all points with $f'(z)\neq 0$. 

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. \hfill 作业3B-2b }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[13.] 
A related property of the mapping is derived by consideration of the modulus $|f'(z_0)|$.

\item[14.] 
We have
\begin{equation*}
\lim\limits_{z\to z_0} \frac{|f(z) - f(z_0)|}{|z-z_0|} = |f'(z_0)|,
\end{equation*}
and this means that any small line segment with one end point at $z_0$ is, in the limit, contracted or expanded in the ratio $|f'(z_0)|$.

\item[15.] 
{\color{red}In other words, the linear change of scale at $z_0$, effected by the transformation $w = f(z)$, is independent of the direction. 
}

\item[16.] 
In general this change of scale will vary from point to point.

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. }

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\begin{enumerate}

\item[17.] 
{\color{red}Conversely, it is clear that both kinds of conformality together imply the existence of $f'(z_0)$. 
}

\item[18.] 
{\color{red}It is less obvious that each kind will separately imply the same result, at least under additional regularity assumptions.
}

\item[19.] 
To be more precise, Let us assume that the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous. 

\item[20.] 
Under this condition the derivative of $w(t) = f(z(t))$ can be expressed in the form 
\begin{equation*}
w'(t_0) = \frac{\partial f}{\partial x}x'(t_0) + \frac{\partial f}{\partial y}y'(t_0)
\end{equation*}
where the partial derivatives are taken at $z_0$. 


\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[21.] 
In terms of $z'(t_0)$ this can be rewritten as
\begin{equation*}
w'(t_0) = \frac{1}{2} \left( \frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y}\right) z'(t_0) + \frac{1}{2} \left( \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right) \overline{z'(t_0)}. 
\end{equation*}

\item[22.] 
If angles are preserved, $\mathrm{arg}\, [w'(t_0)/z'(t_0)]$ must be independent of $\mathrm{arg}\, z'(t_0)$. 

\item[23.] 
The expression
\begin{equation}
\frac{1}{2} \left( \frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y}\right) 
 + \frac{1}{2} \left( \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right) \frac{\overline{z'(t_0)}}{z'(t_0)} 
\label{eq-3}
\end{equation}
must therefore have a constant argument. 

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[24.] 
As $\mathrm{arg}\, z'(t_0)$ is allowed to vary, the point represented by (3) describes a circle having the radius
\begin{equation*}
\frac{1}{2} \left( \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right). 
\end{equation*}

\item[25.] 
The argument cannot be constant on this circle unless its radius vanishes, and hence we must have
\begin{equation}
\frac{\partial f}{\partial x} = - i\frac{\partial f}{\partial y}, 
\label{eq-4}
\end{equation}
which is the complex form of the Cauchy-Riemann equations.

\item[26.] 
{\color{red}Quite similarly, the condition that the change of scale shall be the same in all directions implies that the expression (\ref{eq-3}) has a constant modulus.}

\end{enumerate}

\end{frame}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[27.] 
On a circle the modulus is constant only if the radius vanishes or if the center lies at the origin. 

\item[28.] 
In the first case we obtain (\ref{eq-4}), and in the second case
\begin{equation*}
\frac{\partial f}{\partial x} = i\frac{\partial f}{\partial y}. 
\end{equation*}

\item[29.] 
The last equation expresses the fact that $\overline{f(z)}$ is analytic.

\item[30.] 
A mapping by the conjugate of an analytic function with a nonvanishing derivative is said to be {\color{blue}indirectly conformal}. 

\item[31.] 
It evidently preserves the size but reverses {\color{blue}the sense of angles}.

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[32.] 
If the mapping of $\Omega$ by $w = f(z)$ is {\color{blue}topological}, then the inverse function $z=f^{-1}(w)$ is also analytic. 

\item[33.] 
This follows easily if $f'(z)\neq 0$, for then the derivative of the inverse function must be equal to $1/f'(z)$ at the point $z = f^{-1}(w)$.

\item[34.] 
We shall prove later that $f'(z)$ can never vanish in the case of a topological mapping by an analytic function. 

\end{enumerate}


\vfill 
\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

{\footnotesize a topological mapping: same as a homeomorphism, i.e., a mapping $f:\Omega\to\Omega'$ that is both one-to-one and onto, and both $f$ and $f^{-1}$ are continuous. 
}

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\begin{frame}{2.3. Conformal Mapping. \hfill 作业3B-3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[35.] 
{\color{red}The knowledge that $f'(z_0)\neq 0$ is sufficient to conclude that the mapping is {\color{blue}topological} if it is restricted to a sufficiently small neighborhood of $z_0$. 
}

\item[36.] 
This follows by the {\color{blue}theorem on implicit functions} known from the calculus, for the Jacobian of the functions $u = u(x,y), v = v(x,y)$ at the point $z_0$ is $|f'(z_0)|^2$ and hence $\neq 0$.

\item[37.] 
Later we shall present a simpler proof of this important theorem.

\item[38.] 
But even if $f'(z)\neq 0$ throughout the region $\Omega$, we cannot assert that the mapping of the whole region is necessarily {\color{blue}topological}. 

\item[39.] 
To illustrate what may happen we refer to Fig. 3-1. 

\end{enumerate}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{figure}[ht!]
\centering
\includegraphics[height=0.6\textheight, width=0.65\textwidth]{figure-3-1.png}
%\caption{Fig. 3-1. Doubly covered region }
\end{figure}

\end{frame}

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\begin{frame}{2.3. Conformal Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[40.] 
Here the mappings of the sub-regions $\Omega_1$ and $\Omega_2$ are one to one, but the images overlap. 

\item[41.] 
It is helpful to think of the image of the whole region as a transparent film which partly
covers itself. 

\item[42.] 
This is the simple and fruitful idea used by Riemann when he introduced the {\color{blue}generalized regions} now known as {\color{blue}Riemann surfaces}.

\end{enumerate}

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\begin{frame}{2.4. Length and Area. \hfill 作业3B-4a }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
{\color{red}We have found that under a conformal mapping $f(z)$ the length of an infinitesimal line segment at the point $z$ is multiplied by the factor $|f'(z)|$. 
}

\item[2.] 
Because the distortion is the same in all directions, infinitesimal areas will clearly be multiplied by $|f'(z)|^2$. 

\item[3.] 
Let us put this on a rigorous basis.

\item[4.] 
We know from calculus that the length of a differentiable arc $\gamma$ with the equation $z = z(t) = x(t) + iy(t)$, $a \le t \le b$, is given by
\begin{equation*}
L(\gamma) = \int_a^b \sqrt{x'(t)^2+y'(t)^2}dt = \int_a^b |z'(t)|dt. 
\end{equation*}

\end{enumerate}

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\begin{frame}{2.4. Length and Area. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] 
It is customary to use the shorter notations
\begin{equation}
L(\gamma) = \int_{\gamma} |dz|, \hspace{0.5cm}
L(\gamma') = \int_{\gamma} |f'(z)||dz|. 
\label{eq-5}
\end{equation}

\item[6.] 
Observe that in complex notation the calculus symbol $ds$ for integration with respect to arc length is replaced by $|dz|$.

\item[7.] 
Now let $E$ be a point set in the plane whose area 
\begin{equation*}
A(E) = \iint_E dxdy
\end{equation*}
can be evaluated as a double Riemann integral. 

\end{enumerate}

\end{frame}

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\begin{frame}{2.4. Length and Area. \hfill 作业3B-4b }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[8.] 
{\color{red}
If $f(z) = u(x,y) + iv(x,y)$ is a {\color{blue}bijective} differentiable mapping, then by the rule for changing integration variables the area of the image $E' = f(E)$ is given by 
\begin{equation*}
A(E') = \iint_E |u_xv_y - u_yv_x|dxdy.
\end{equation*}
}

\item[9.] 
But if $f(z)$ is a conformal mapping of an open set containing $E$, then
$u_xv_y - u_yv_x = |f'(z)|^2$ by virtue of the Cauchy-Riemann equations, and we obtain
\begin{equation}
A(E') = \iint_E |f'(z)|^2dxdy.
\label{eq-6}
\end{equation}

\item[10.] 
The formulas (\ref{eq-5}) and (\ref{eq-6}) have important applications in the part of complex analysis that is frequently referred to as geometric function theory.


\end{enumerate}

\end{frame}

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